Nonlinear Schrödinger equations have been broadly investigated
in many aspects, such as existence of solitary waves, concentration
and multi-bump phenomena for semiclassical states (see e.g.
[1]}, [2]} and the references therein). The study on
system of Schrödinger equations began quite recently. Mathematical
work on systems with the nonlinearly coupling terms (e.g. the term
\(\sum _{i\ne j}^N u_i\)  in (REF ) being replaced by
\(u_j\sum _{i\ne j}^N u_i^2\)  ) has been studied extensively in
recent years, for example,
[3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]} and
references therein, where phase separation or synchronization has
been proved in several cases.
